Bikes are FAR cheaper to buy in the USA than Latin America.
Texas is good to buy as is California, Arizona, Oregon.

There are 10 times as many good used bikes in the USA than anywhere in Mex, Cent. or South America.

With $2000 usd you can do OK, but $2500 would get you something better.
Craig's list is your friend. Be patient, shop around. Have cash ready when you find a good deal. Try to get seller to help you register the bike in your name and help with transfer. (not hard)

You must have the bike's title in your name to cross borders. Not a big deal. You may have to purchase US insurance (or not) legally required.

Once in Mexico, NO USA insurance is valid there, so you should buy Mexican insurance (although not mandatory, but a very good idea) You can buy Mex. insurance for any amount of time and you can add time as you like.

There are tons of threads here on HUBB about all this. Read up!

I'd look at 250's, 400's or 650's, street legal bike is best. Most prefer some sort of Japanese made dual sport bike as they are TOUGH, versatile and reliable. They pack up well and very easy to service and maintain. They use common tire sizes and parts may be more available down South.

Here are some more common, low priced bikes you might find out there
(listed dual sport bikes only, USA bikes only):

HONDA
XR250L, XL250R, CRF230L*, (XR250R or XR400R* with Baja street kit), XR650L

Suzuki
DR200, DRZ400S* or E, DR650SE* (buy post '97 only)

Yamaha
XT225 Serow, XT250, XT600R (rare)

Kawasaki
225 Super Sherpa*, KLR250, KLX250S, KLX400S* (same as Suzuki DRZ400S) KLX650 (rare). KLR650*.

* indicates recommended

KLR650's are quite common and cheap-ish. Try to get one that's set up for travel. DR650 is the bike I own, a good one but a bit more $$ than KLR. Good 250's may be hard to find and a bit too much money but if lots of off road riding planned, then may be preferable.

Many thanks for the info guys!

Mollydog that was very informative, appreciate it.

Had a look at the houston craigslist which definitely has more than new orleans. That XR option doesn't sound bad at all Plenty more to read but at least now I've got a better idea of the where, so have to check the how.

safe travels to all

If you can find a Qlink XF/xp 200 or a SSR XF 200 or 250, these are Chinese made but made in a factory with some kind of deal with Suzuki, Qingqi. They are found all over Central and South America (under different brand names) so parts and/or service would be easy to find. The advantage is they are brand new for near your budget as opposed to finding and sorting out someone's used bike. Many Suzuki engine parts interchange with the Qingqi engine, too. Put in Qingqi in the search feature and I am sure a few threads will pop up here on Horizons. I mention them simply because of their low cost and above average (for Chinese) quality. Plus, they are found throughout the US or mail ordered to you.